Marked groups with isomorphic Cayley graphs but different Borel combinatorics

Weilacher, Felix

doi:10.4064/fm863-2-2020

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Measure Chromatic Numbers1

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2022

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“…Thus, the bound in Theorem 4.1 is indeed sharp. Similarly, alongside Theorem 1.6, Weilacher [9] proves that there are marked groups with isomorphic Cayley graphs can have measure chromatic numbers which differ by one but notes that it is open whether or not these numbers can differ by more than one. By Proposition 4.2, we have resolved this as well.…”

Section: Measure Chromatic Numbersmentioning

confidence: 91%

Descriptive chromatic numbers of locally finite and everywhere two-ended graphs

Weilacher

2022

Groups Geom. Dyn.

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We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include "Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?" and "Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?" We also provide a new bound for Borel chromatic numbers of graphs whose connected components all have two ends.

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Section: Measure Chromatic Numbersmentioning

confidence: 91%